# CHAPTER 12

# SOLVING LINEAR EQUATIONS

# 12.1 Least-Squares Analysis

Consider a system of linear equations

where *A*^{m×n}, *b*^{m}, *m* ≥ *n*, and rank ** A** =

*n*. Note that the number of unknowns,

*n*, is no larger than the number of equations,

*m*. If

**does not belong to the range of**

*b***, that is, if**

*A***∉ (**

*b***), then this system of equations is said to be**

*A**inconsistent*or

*overdetermined.*In this case there is no solution to the above set of equations. Our goal then is to find the vector (or vectors)

**minimizing ||**

*x***−**

*Ax***||**

*b*^{2}. This problem is a special case of the nonlinear least-squares problem discussed in Section 9.4.

Let ** x*** be a vector that minimizes ||

**−**

*Ax***||**

*b*^{2}; that is, for all

*x*^{n},

We refer to the vector ** x*** as ...

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